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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 20, Pages 1047-1056

Compact and extremally disconnected spaces

Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

Received 22 August 2002

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Viglino defined a Hausdorff topological space to be C-compact if each closed subset of the space is an H-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is an S-set in the sense of Dickman and Krystock. Such spaces are called C-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterize C-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown that C-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class of C-s-compact spaces is properly contained in the class of C-compact spaces as well as in the class of S-closed spaces of Thompson. In general, a compact space need not be C-s-compact. The product of two C-s-compact spaces need not be C-s-compact.